In this post, I’m going to calculate the two points of intersection, a and b, you see in Figure 1.

The “lens” area *c* is known as a Vesica Piscis (or fish liver, in Latin). We can use a formula seen here to help us in our computations.

length of the radical axis *h* (this is the height of the lens) equals 1/2 the radius of our circle times the square root of 3

Plugging in the numbers…

h = 1.732 x 100 = 173.2px

173.2 / 2 = 86.6px

a = 200 – 86.6 = 113.4px (the value of the y coordinate)

thus point *a* is located at (250, 113.4)

which can be verified in Inkscape (see Figure 2, the blue oval).

It’s also possible to calculate the coordinates of point *b.*

Notice we have 2 right-sided triangles connected at point *a* in Figure 1:

50 (opposite side) in px units) / 100 (hypotenuse) = 0.5 radians

(-1) sine of 0.5 radians = ∠30°

We can now calculate the coordinates of point *b,* using the tan(θ) = **O**pposite / **A**djacent formula.

Calculation:

Opposite side = 113.4 – 100 (y coordinate of rectangle’s top border) = 13.4 px

tan 30 degrees = X / 13.4

X = 0.577 * 13.4 = 7.7318

x-coordinate = 250 – 7.7318 = 242.2682

As you can see from the SVG code snippet below, this is the same value as the one I arrived through trial and error using Inkscape and a mouse pointer.

<line x1="242" y1="100" x2="358.2" y2="300" />

QED

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